2. Symmetry and Mechanism of Multiferroicity in Frustrated Magnets 黃迪靖 and 牟中瑜 Resonant soft x-ray scattering Ginzburg-Landau approach

3. 1 Collaborators Soft x-ray scattering: J. Okamoto, H.-J. Lin, and C. T. Chen (NSRRC, Taiwan) K. S. Chao (National Chiao-Tung University, Taiwan) TbMn 2 O 5 : S. Park, S. W. Cheong (Rutgers University, USA) Acknowledgement L. L. Lee, H. W. Fu, and S. C. Chung (NSRRC, Taiwan) S. Ishihara (Tohoku University, Japan) Y. Tokura (Univ. of Tokyo, Japan) C. H. Chen (University of Taiwan) T. K. Lee (Academia Sinica)

4. 2 Introduction Resonant soft x-ray scattering Ginzburg-Landau approach

5. 3 Magnetism: ordering of spins Ferroelectricity: polar arrangement of charges strain  E T  E PZT “Fire”(-Pyro)electricity “Pressure”(-piezo) electricity

6. 4 (Ferro)magnetism vs. (Ferro)electricity Perovskite structure (La,Sr)MnO 3 : spins from : 3d 3 or 3d 4 BaTiO 3 : polarization from cation/anion paired diploes O -2 Ti +4 Magnetic moment: - Ba +2 0.10 Å 0.05 Å 0.04 Å + + Ti 3d 0 O 2p 2 unfilled d bands impurities inversion symmetry broken

7. 5 BiMnO 3, BiFeO 3, Pb(Fe 2/3 W 1/3 )O 3 : 6s 2 lone pairs  off-center distortion  polar behavior uncorrelated with magnetism ? Mechanism of ferroelectricity PbTiO 3 : Pb-O covalent bond cubic 800 K tetragonal 300 K Pb-O planeTi-O plane Pb O Kuroiwa et al, PRL87 217601 (2001)

8. 6 Induction of magnetization by an electric field; induction of polarization by a magnetic field. - first presumed to exist by Pierre Curie in 1894 on the basis of symmetry considerations However, the effects are typically too small to be useful in applications! Magnetoelectric effect Materials exhibiting ME effect: Cr 2 O 3 BiMnO 3 BiFeO 3 ….. M. Fiebig, J. Phys. D: Appl. Phys 38, R123 (2005)

9. Three Scenarios

10. Lawes et al., PRL 91, 257208, 2003 Hexagonal RMnO 3 : T C : 570-900 K, T N =70-130 K BiFeO 3 : ferroelectric, ferroelastic, and weakly ferromagnetic; rhombohedrally distorted; T C =1100 K, T N = 650 K

11. c N N

12. 10 Recently discovery in the coexistence and gigantic coupling of antiferromagnetism and ferroelectricity in frustrated spin systems such RMnO 3 and RMn 2 O 5 (R=Tb, Ho, …) revived interest in “multiferroic” systems

13. Recent Discoveries Frustrated magnetic systems. The magnetic phases are complicated; incommensurate AF orders seem to be common. Strong coupling between ferroelectricty and magnetism. TbMnO 3, Nature 426, 55, (2003); PRL 95, 087206 (2005). RMn 2 O 5, Nature 429, 392 (2004) (Tb); PRL 96, 067601 (2006) (Y).

14. 12 *Can the ME be enhanced by the internal fields? Multiferroicity: coexistence of magnetism and ferroelectricity with cross coupling How to enhance the coupling?

15. 13 Geometric ferroelectrics: hexagonal RMnO 3 BaNi(Mn,Co,Fe)F 4 For example: YMnO 3 : A-type AF, lacking lone pairs, bulking of MnO 5 pyramids & displacement of Y  polarization Origin of ferroelectricity van Aken et al., Nature Materials 3, 164, (2004) Y MnO 5

16. 14 Site-centered charge order Electronic ferroelectrics: Combination of bond-centered and site-centered charge order Efremov et al., Nature Materials 3, 853, (2004) Bond-centered charge order Ferroelectric intermediate state O TM

17. 15 Magnetic ferroelectrics: (frustrated spin systems) Dzyaloshinskii-Moriya interaction ( H =D. S 1 X S 2 ) spin current “electromagnon” (spin waves excited by ac E fields) Geometric ferroelectrics: hexagonal RMnO 3 BaNi(Mn, Co, Fe)F 4 Origin of ferroelectricity Sergienko and Dagotto, Phys. Rev. B 73, 094434 (2006) Katsura, Nagaosa and Baltasky, Phys. Rev. Lett. 95, 057205 (2006) Electronic ferroelectrics: Combination of bond-centered and site-centered charge order Efremov et al., Nature Materials 3, 853, (2004) Pimenov et al. Nature Phys (2006)

18. 16 Nature, 426, 55 (2003) TbMnO 3 incommensurate AF order  ME effect

19. 17 T=35 K T=15 K T N =42 K T C =27 K TbMnO 3 Kenzelmann et al., PRL 95, 087206 (2005) IC sinusoidally modulated collinear magnetic order, inversion symmetric IC noncollinear magnetic order, inversion symmetry broken

20. 18 3 transitions on cooling. Magnetic field induces a sign reversal of the electric polarization. TbMn 2 O 5 Nature, 429, 392 (2004)

21. 19 Difficulty in microscopic measurement L. C. Chapon et al. Phys. Rev. Lett. 93, 177402, 2004 Scattering amplitude accumulates microscopic effects and is macroscopic:

22. 20 Nature of the transition N. Hur et al. Phys. Rev. Lett. 93, 107207, 2004

23. 21 Tb O Mn 4+ O 6 octahedron q P orthorhombic structure (a  b  c,  =  =  = 90˚) AFM insulator (T N =42 K ) magnetization in the ab plane AFM square lattice with asymmetrical next-nearest- neighbor interactions, i.e., geometrically frustrated Spontaneous polarization P // b q  P Mn 3+ O 5 pyramid TbMn 2 O 5 Tb 3+ Mn 4+, Mn 3+ O 2- Chapon et al, PRL (2004) Blake et al, PRB (2005)

24. 22 Blake et al., PRB (2005) 20° ab plane 6°6° 31°17° TbMn 2 O 5 Mn 4+ O 6 Mn 3+ O 5 Tb 3+ O 2- The spins lie in the ab plane. Within the ab plane, two zigzag chains of AFM-coupled nearest-neighbor Mn 4+ and Mn 3+ run in a direction parallel to the a axis. orthorhombic structure a=7.3233 Å, b=8.5205 Å c=5.6601 Å

25. 23 qxqx qzqz 0 20 40 60 Temperature (K) 0.25 0.30 0.50 0.55 Chapon et al, PRL (2004) (½ 0 ¼) commensurate incommensurate 33 incommensurate 24 42 Neutron diffraction: complex spin structure AFM, T N = 42 K, q = (q x 0 q z ) Mechanism of ME effect ?

26. 24 qxqx qzqz 0 20 40 60 Temperature (K) 0.25 0.30 0.50 0.55 Chapon et al, PRL (2004) (½ 0 ¼) commensurate incommensurate 33 incommensurate 24 42 Neutron diffraction: complex spin structure Kobayashi et al, JPSJ(2004) 37 24 42 AFM, T N = 42 K modulation vector q = (q x 0 q z )

27. 25 Issues: -What is the underlying mechanism of the gigantic ME effect? -Is a spiral-spin configuration necessary? -Can collinear spins lead to a polarization?

28. 26 Introduction Resonant soft x-ray scattering Ginzburg-Landau approach

29. 27 Elastic x-ray scattering scattering form factor momentum transfer A volume element at will contribute an amount to the scattering field with a phase factor. Fourier transform of charge distribution. Bragg condition: q = modulation vector of charge, spin, or orbital order elastic scattering Fourier transform of spin distribution. detectable?

30. 28 X-ray scattering : electron density charge scatteringmagnetic-moment scattering Non-resonant X-ray magnetic scattering is very weak. (for ~ 600 eV)

31. 29 Resonant X-ray magnetic scattering electric dipole transitions F 1,1  F 1,-1 scattering amplitudes enhanced Hannon et al., PRL(1988) 2p 3/2 3d 2p 1/2 (F 1,±1 => S q )

32. 30 beamline 4-m long elliptically polarized undulator UHV, two-circle diffractometer detector Soft X-ray Scattering Set-up at NSRRC, Taiwan H. J. Lin & C. T. Chen et al.

33. 31 Soft x-ray scattering of TbMn 2 O 5 3d 2p 3/2 Mn cm TbMn 2 O 5 single crystal h = 637.7 eV E  b T = 30 K q [100] [001] [010]

34. 32 (½ 0 ¼) commensurate incommensurate Coexistence of ICM and CM AF order 3d 2p 3/2 Mn 640 eV

35. 33 qxqx qzqz Kobayashi et al, JPSJ(2004) CM 3d 2p 3/2 Mn q z = 1/4 IC

36. 34 AF transitions closely resembles ferroelectric transitions ICM AF order   ICM CM

37. 35 q Antiferromagnetism and ferroelectrics are strongly coupled.

38. 36 Introduction Resonant soft x-ray scattering Ginzburg-Landau approach

39. 37 Magnetoelectric Effect (ME) *Induction of M by E, induction of P by H *Can the ME be enhanced by the internal fields?

40. 38 Symmetry consideration Two important symmetries:

41. 39 Symmetry properties

42. 40 Inversion symmetry broken in magnetic phases

43. 41 Blake et al., PRB (2005) 20° ab plane 6°6° 31°17° TbMn 2 O 5 Mn 4+ O 6 Mn 3+ O 5 Tb 3+ O 2- The spins lie in the ab plane. Within the ab plane, two zigzag chains of AFM-coupled nearest-neighbor Mn 4+ and Mn 3+ run in a direction parallel to the a axis. a b c a d a b c d d c b

44. 42 Inversion symmetric magnetic phase

45. 43 Inversion Symmetry versus collinearity

46. 44 Inversion symmetric broken - spiral Example of non-common phases

47. 45 Other spirals

48. 46 Inversion symmetric broken - collinear

49. 47 Coupling to polarization Inversion Symmetry in Magnetic Phase Odd orders of P involved: Even orders of P involved:

50. 48 Magnetic induced polarization with inversion symmetry broken Lowest order -- the existence of internal fields

51. 49 Symmetry Constraints

52. 50 not consistent with expts.

53. 51 Electric Polarization Change and Reversal Strong magnetic field along a or b axis:

54. 52 Nature 429, 392, 2004 Consistent with our result but why it does not happen for b-axis? Possibility: a-axis is an easy axis, and it is very hard to reverse S b P must be odd in S a !

55. 53 Direct comparison with expt commensurate+incommensur ate incommensurate

56. 54 Nature of the transition collinear or Noncollinear and Inversion-symmetry broken Inversion-symmetric

57. 55 Nature of the transition N. Hur et al. Phys. Rev. Lett. 93, 107207, 2004

58. 56 Effect of External Fields - the dielectric constant Nature 429, 392, 2004

59. 57 Expansion of Free Energy

60. 58 Change of exchange energy

61. 59 Dielectric Anomaly (Lawes et al., RPL 25, 257208, 2003) Small step

62. 60 Large Step at IC transition

63. 61 Wavenumber change of IC moments * Lock-in transition * Change of q

64. 62 Connection with Microscopic Theories Sergienko and Dagotto, Phys. Rev. B 73, 094434 (2006) Dzyaloshinskii-Moriya interaction with atomic displacement

65. 63 Katsura, Nagaosa and Baltasky, Phys. Rev. Lett. 95, 057205 (2006) Theory of Katsura, Nagaosa and Baltasky 1. Polarization without atomic displacement Efremov et al., Nature Materials 3, 853, 2004 2.

66. 64 How to induce polarization without involving atomic displacement? Essential Physics: Motion of magnetic moments induces electric dipoles! – the intrinsic Aharonov-Casher Effect

67. 65 Katsura, Nagaosa and Baltasky : Motion of magnetic moments= spin current

68. 66 Aharonov-Casher Effect in condensed matter (Meier and Loss, PRL 90, 167204,2003)

69. 67

70. 68 Conclusion

71. 69 Conclusion For magnetic phases with inversion symmetry broken TbMn2O5 Consistent well with our expts on TbMn2O5 *Response to the internal field: *Response to the external electric field: Consistent with magneto-elastic effect on exchange energy

72. 70 Induction of magnetization by an electric field; induction of polarization by a magnetic field. - first presumed to exist by Pierre Curie in 1894 on the basis of symmetry considerations Magnetoelectric effect M. Fiebig, J. Phys. D: Appl. Phys 38, R123 (2005)

73. 71 Old Examples Ferromagnetic, ferroelectric, and ferroelastic at Tc=61.5K

74. 72 Antiferromagnetic (spiral) & ferroelectric at Tc=28K Ferroelastic at high temperature but Antiferromagnetic at Tc=25-70K